-
Notifications
You must be signed in to change notification settings - Fork 387
/
kendall.py
409 lines (338 loc) · 14.8 KB
/
kendall.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
# Copyright The Lightning team.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from typing import List, Optional, Tuple, Union
import torch
from torch import Tensor
from typing_extensions import Literal
from torchmetrics.functional.regression.utils import _check_data_shape_to_num_outputs
from torchmetrics.utilities.checks import _check_same_shape
from torchmetrics.utilities.data import _bincount, _cumsum, dim_zero_cat
from torchmetrics.utilities.enums import EnumStr
class _MetricVariant(EnumStr):
"""Enumerate for metric variants."""
A = "a"
B = "b"
C = "c"
@staticmethod
def _name() -> str:
return "variant"
class _TestAlternative(EnumStr):
"""Enumerate for test alternative options."""
TWO_SIDED = "two-sided"
LESS = "less"
GREATER = "greater"
@staticmethod
def _name() -> str:
return "alternative"
def _sort_on_first_sequence(x: Tensor, y: Tensor) -> Tuple[Tensor, Tensor]:
"""Sort sequences in an ascent order according to the sequence ``x``."""
# We need to clone `y` tensor not to change an object in memory
y = torch.clone(y)
x, y = x.T, y.T
x, perm = x.sort()
for i in range(x.shape[0]):
y[i] = y[i][perm[i]]
return x.T, y.T
def _concordant_element_sum(x: Tensor, y: Tensor, i: int) -> Tensor:
"""Count a total number of concordant pairs in a single sequence."""
return torch.logical_and(x[i] < x[(i + 1) :], y[i] < y[(i + 1) :]).sum(0).unsqueeze(0)
def _count_concordant_pairs(preds: Tensor, target: Tensor) -> Tensor:
"""Count a total number of concordant pairs in given sequences."""
return torch.cat([_concordant_element_sum(preds, target, i) for i in range(preds.shape[0])]).sum(0)
def _discordant_element_sum(x: Tensor, y: Tensor, i: int) -> Tensor:
"""Count a total number of discordant pairs in a single sequences."""
return (
torch.logical_or(
torch.logical_and(x[i] > x[(i + 1) :], y[i] < y[(i + 1) :]),
torch.logical_and(x[i] < x[(i + 1) :], y[i] > y[(i + 1) :]),
)
.sum(0)
.unsqueeze(0)
)
def _count_discordant_pairs(preds: Tensor, target: Tensor) -> Tensor:
"""Count a total number of discordant pairs in given sequences."""
return torch.cat([_discordant_element_sum(preds, target, i) for i in range(preds.shape[0])]).sum(0)
def _convert_sequence_to_dense_rank(x: Tensor, sort: bool = False) -> Tensor:
"""Convert a sequence to the rank tensor."""
# Sort if a sequence has not been sorted before
if sort:
x = x.sort(dim=0).values
_ones = torch.zeros(1, x.shape[1], dtype=torch.int32, device=x.device)
return _cumsum(torch.cat([_ones, (x[1:] != x[:-1]).int()], dim=0), dim=0)
def _get_ties(x: Tensor) -> Tuple[Tensor, Tensor, Tensor]:
"""Get a total number of ties and staistics for p-value calculation for a given sequence."""
ties = torch.zeros(x.shape[1], dtype=x.dtype, device=x.device)
ties_p1 = torch.zeros(x.shape[1], dtype=x.dtype, device=x.device)
ties_p2 = torch.zeros(x.shape[1], dtype=x.dtype, device=x.device)
for dim in range(x.shape[1]):
n_ties = _bincount(x[:, dim])
n_ties = n_ties[n_ties > 1]
ties[dim] = (n_ties * (n_ties - 1) // 2).sum()
ties_p1[dim] = (n_ties * (n_ties - 1.0) * (n_ties - 2)).sum()
ties_p2[dim] = (n_ties * (n_ties - 1.0) * (2 * n_ties + 5)).sum()
return ties, ties_p1, ties_p2
def _get_metric_metadata(
preds: Tensor, target: Tensor, variant: _MetricVariant
) -> Tuple[
Tensor,
Tensor,
Optional[Tensor],
Optional[Tensor],
Optional[Tensor],
Optional[Tensor],
Optional[Tensor],
Optional[Tensor],
Tensor,
]:
"""Obtain statistics to calculate metric value."""
preds, target = _sort_on_first_sequence(preds, target)
concordant_pairs = _count_concordant_pairs(preds, target)
discordant_pairs = _count_discordant_pairs(preds, target)
n_total = torch.tensor(preds.shape[0], device=preds.device)
preds_ties = target_ties = None
preds_ties_p1 = preds_ties_p2 = target_ties_p1 = target_ties_p2 = None
if variant != _MetricVariant.A:
preds = _convert_sequence_to_dense_rank(preds)
target = _convert_sequence_to_dense_rank(target, sort=True)
preds_ties, preds_ties_p1, preds_ties_p2 = _get_ties(preds)
target_ties, target_ties_p1, target_ties_p2 = _get_ties(target)
return (
concordant_pairs,
discordant_pairs,
preds_ties,
preds_ties_p1,
preds_ties_p2,
target_ties,
target_ties_p1,
target_ties_p2,
n_total,
)
def _calculate_tau(
preds: Tensor,
target: Tensor,
concordant_pairs: Tensor,
discordant_pairs: Tensor,
con_min_dis_pairs: Tensor,
n_total: Tensor,
preds_ties: Optional[Tensor],
target_ties: Optional[Tensor],
variant: _MetricVariant,
) -> Tensor:
"""Calculate Kendall's tau from metric metadata."""
if variant == _MetricVariant.A:
return con_min_dis_pairs / (concordant_pairs + discordant_pairs)
if variant == _MetricVariant.B:
total_combinations: Tensor = n_total * (n_total - 1) // 2
denominator = (total_combinations - preds_ties) * (total_combinations - target_ties)
return con_min_dis_pairs / torch.sqrt(denominator)
preds_unique = torch.tensor([len(p.unique()) for p in preds.T], dtype=preds.dtype, device=preds.device)
target_unique = torch.tensor([len(t.unique()) for t in target.T], dtype=target.dtype, device=target.device)
min_classes = torch.minimum(preds_unique, target_unique)
return 2 * con_min_dis_pairs / ((min_classes - 1) / min_classes * n_total**2)
def _get_p_value_for_t_value_from_dist(t_value: Tensor) -> Tensor:
"""Obtain p-value for a given Tensor of t-values. Handle ``nan`` which cannot be passed into torch distributions.
When t-value is ``nan``, a resulted p-value should be alson ``nan``.
"""
device = t_value
normal_dist = torch.distributions.normal.Normal(torch.tensor([0.0]).to(device), torch.tensor([1.0]).to(device))
is_nan = t_value.isnan()
t_value = t_value.nan_to_num()
p_value = normal_dist.cdf(t_value)
return p_value.where(~is_nan, torch.tensor(float("nan"), dtype=p_value.dtype, device=p_value.device))
def _calculate_p_value(
con_min_dis_pairs: Tensor,
n_total: Tensor,
preds_ties: Optional[Tensor],
preds_ties_p1: Optional[Tensor],
preds_ties_p2: Optional[Tensor],
target_ties: Optional[Tensor],
target_ties_p1: Optional[Tensor],
target_ties_p2: Optional[Tensor],
variant: _MetricVariant,
alternative: Optional[_TestAlternative],
) -> Tensor:
"""Calculate p-value for Kendall's tau from metric metadata."""
t_value_denominator_base = n_total * (n_total - 1) * (2 * n_total + 5)
if variant == _MetricVariant.A:
t_value = 3 * con_min_dis_pairs / torch.sqrt(t_value_denominator_base / 2)
else:
m = n_total * (n_total - 1)
t_value_denominator: Tensor = (t_value_denominator_base - preds_ties_p2 - target_ties_p2) / 18
t_value_denominator += (2 * preds_ties * target_ties) / m # type: ignore
t_value_denominator += preds_ties_p1 * target_ties_p1 / (9 * m * (n_total - 2)) # type: ignore
t_value = con_min_dis_pairs / torch.sqrt(t_value_denominator)
if alternative == _TestAlternative.TWO_SIDED:
t_value = torch.abs(t_value)
if alternative in [_TestAlternative.TWO_SIDED, _TestAlternative.GREATER]:
t_value *= -1
p_value = _get_p_value_for_t_value_from_dist(t_value)
if alternative == _TestAlternative.TWO_SIDED:
p_value *= 2
return p_value
def _kendall_corrcoef_update(
preds: Tensor,
target: Tensor,
concat_preds: Optional[List[Tensor]] = None,
concat_target: Optional[List[Tensor]] = None,
num_outputs: int = 1,
) -> Tuple[List[Tensor], List[Tensor]]:
"""Update variables required to compute Kendall rank correlation coefficient.
Args:
preds: Sequence of data
target: Sequence of data
concat_preds: List of batches of preds sequence to be concatenated
concat_target: List of batches of target sequence to be concatenated
num_outputs: Number of outputs in multioutput setting
Raises:
RuntimeError: If ``preds`` and ``target`` do not have the same shape
"""
concat_preds = concat_preds or []
concat_target = concat_target or []
# Data checking
_check_same_shape(preds, target)
_check_data_shape_to_num_outputs(preds, target, num_outputs)
if num_outputs == 1:
preds = preds.unsqueeze(1)
target = target.unsqueeze(1)
concat_preds.append(preds)
concat_target.append(target)
return concat_preds, concat_target
def _kendall_corrcoef_compute(
preds: Tensor,
target: Tensor,
variant: _MetricVariant,
alternative: Optional[_TestAlternative] = None,
) -> Tuple[Tensor, Optional[Tensor]]:
"""Compute Kendall rank correlation coefficient, and optionally p-value of corresponding statistical test.
Args:
Args:
preds: Sequence of data
target: Sequence of data
variant: Indication of which variant of Kendall's tau to be used
alternative: Alternative hypothesis for for t-test. Possible values:
- 'two-sided': the rank correlation is nonzero
- 'less': the rank correlation is negative (less than zero)
- 'greater': the rank correlation is positive (greater than zero)
"""
(
concordant_pairs,
discordant_pairs,
preds_ties,
preds_ties_p1,
preds_ties_p2,
target_ties,
target_ties_p1,
target_ties_p2,
n_total,
) = _get_metric_metadata(preds, target, variant)
con_min_dis_pairs = concordant_pairs - discordant_pairs
tau = _calculate_tau(
preds, target, concordant_pairs, discordant_pairs, con_min_dis_pairs, n_total, preds_ties, target_ties, variant
)
p_value = (
_calculate_p_value(
con_min_dis_pairs,
n_total,
preds_ties,
preds_ties_p1,
preds_ties_p2,
target_ties,
target_ties_p1,
target_ties_p2,
variant,
alternative,
)
if alternative
else None
)
# Squeeze tensor if num_outputs=1
if tau.shape[0] == 1:
tau = tau.squeeze()
p_value = p_value.squeeze() if p_value is not None else None
return tau.clamp(-1, 1), p_value
def kendall_rank_corrcoef(
preds: Tensor,
target: Tensor,
variant: Literal["a", "b", "c"] = "b",
t_test: bool = False,
alternative: Optional[Literal["two-sided", "less", "greater"]] = "two-sided",
) -> Union[Tensor, Tuple[Tensor, Tensor]]:
r"""Compute `Kendall Rank Correlation Coefficient`_.
.. math::
tau_a = \frac{C - D}{C + D}
where :math:`C` represents concordant pairs, :math:`D` stands for discordant pairs.
.. math::
tau_b = \frac{C - D}{\sqrt{(C + D + T_{preds}) * (C + D + T_{target})}}
where :math:`C` represents concordant pairs, :math:`D` stands for discordant pairs and :math:`T` represents
a total number of ties.
.. math::
tau_c = 2 * \frac{C - D}{n^2 * \frac{m - 1}{m}}
where :math:`C` represents concordant pairs, :math:`D` stands for discordant pairs, :math:`n` is a total number
of observations and :math:`m` is a ``min`` of unique values in ``preds`` and ``target`` sequence.
Definitions according to Definition according to `The Treatment of Ties in Ranking Problems`_.
Args:
preds: Sequence of data of either shape ``(N,)`` or ``(N,d)``
target: Sequence of data of either shape ``(N,)`` or ``(N,d)``
variant: Indication of which variant of Kendall's tau to be used
t_test: Indication whether to run t-test
alternative: Alternative hypothesis for t-test. Possible values:
- 'two-sided': the rank correlation is nonzero
- 'less': the rank correlation is negative (less than zero)
- 'greater': the rank correlation is positive (greater than zero)
Return:
Correlation tau statistic
(Optional) p-value of corresponding statistical test (asymptotic)
Raises:
ValueError: If ``t_test`` is not of a type bool
ValueError: If ``t_test=True`` and ``alternative=None``
Example (single output regression):
>>> from torchmetrics.functional.regression import kendall_rank_corrcoef
>>> preds = torch.tensor([2.5, 0.0, 2, 8])
>>> target = torch.tensor([3, -0.5, 2, 1])
>>> kendall_rank_corrcoef(preds, target)
tensor(0.3333)
Example (multi output regression):
>>> from torchmetrics.functional.regression import kendall_rank_corrcoef
>>> preds = torch.tensor([[2.5, 0.0], [2, 8]])
>>> target = torch.tensor([[3, -0.5], [2, 1]])
>>> kendall_rank_corrcoef(preds, target)
tensor([1., 1.])
Example (single output regression with t-test)
>>> from torchmetrics.functional.regression import kendall_rank_corrcoef
>>> preds = torch.tensor([2.5, 0.0, 2, 8])
>>> target = torch.tensor([3, -0.5, 2, 1])
>>> kendall_rank_corrcoef(preds, target, t_test=True, alternative='two-sided')
(tensor(0.3333), tensor(0.4969))
Example (multi output regression with t-test):
>>> from torchmetrics.functional.regression import kendall_rank_corrcoef
>>> preds = torch.tensor([[2.5, 0.0], [2, 8]])
>>> target = torch.tensor([[3, -0.5], [2, 1]])
>>> kendall_rank_corrcoef(preds, target, t_test=True, alternative='two-sided')
(tensor([1., 1.]), tensor([nan, nan]))
"""
if not isinstance(t_test, bool):
raise ValueError(f"Argument `t_test` is expected to be of a type `bool`, but got {type(t_test)}.")
if t_test and alternative is None:
raise ValueError("Argument `alternative` is required if `t_test=True` but got `None`.")
_variant = _MetricVariant.from_str(str(variant))
_alternative = _TestAlternative.from_str(str(alternative)) if t_test else None
_preds, _target = _kendall_corrcoef_update(
preds, target, [], [], num_outputs=1 if preds.ndim == 1 else preds.shape[-1]
)
tau, p_value = _kendall_corrcoef_compute(
dim_zero_cat(_preds), dim_zero_cat(_target), _variant, _alternative # type: ignore[arg-type] # todo
)
if p_value is not None:
return tau, p_value
return tau